Undergrad Instability of the Solar System

[Silviu]

Hello! I read in some books that the behavior of the Solar system can't be predicted indefinitely (up to several hundreds of million of years). Is this a numerical limitation, in trying to simulate it or is it some physics responsible for it? (I guess that this assumes that no other bodies interfere with the system, otherwise you would need to take into account the whole light cone for 100 million years and I think this is not doable with current technology) Thank you!

 

[jedishrfu]

Simulations are often approximations to a real system and don't take into account every facet of the real system.

In the solar system case, over a long period of time you will need to factor in how one planet affects another and the then it becomes an n-body problem which is extremely impractical to impossible to model perfectly.

https://en.wikipedia.org/wiki/Stability_of_the_Solar_System

I've seen simulations of three body problem which easily become chaotic as they run. Error is introduced just from repeated summing which either adds energy to the system or takes it away. Often modelers try to use algorithms that while not perfect periodically add error and then remove it (error is manifest as energy in the run) so that over a long time the run is stable.



https://en.wikipedia.org/wiki/Three-body_problem

https://en.wikipedia.org/wiki/N-body_problem

https://en.wikipedia.org/wiki/N-body_simulation

 

[Silviu]

Simulations are often approximations to a real system and don't take into account every facet of the real system.

In the solar system case, over a long period of time you will need to factor in how one planet affects another and the then it becomes an n-body problem which is extremely impractical to impossible to model perfectly.

https://en.wikipedia.org/wiki/Stability_of_the_Solar_System

I've seen simulations of three body problem which easily become chaotic as they run. Error is introduced just from repeated summing which either adds energy to the system or takes it away. Often modelers try to use algorithms that while not perfect periodically add error and then remove it (error is manifest as energy in the run) so that over a long time the run is stable.



https://en.wikipedia.org/wiki/Three-body_problem

https://en.wikipedia.org/wiki/N-body_problem

https://en.wikipedia.org/wiki/N-body_simulation

But in the end, this is a numerical problem. The Solar System motion is deterministic, it is just a limitation of our computers (like round-off errors), right?

 

[jedishrfu]

We don't know if the solar system is stable for the long term, only that it has been for some time and that it will be for some time. The solar system is not a stable system over the long run.

Chaotic systems can't be predicted because that computer has a finite precision in its numbers and no matter how many decimals we choose to save, its in these lost decimal values that chaos springs up. Saturn's moons have chaotic orbits and a couple of moons have a dance where they switch orbits as they cross.

Basically small changes in one state get magnified in future states until the simulation no longer matches reality. We say the system is non-linear.

https://en.wikipedia.org/wiki/Butterfly_effect

 

[Silviu]

We don't know if the solar system is stable for the long term, only that it has been for some time and that it will be for some time.

Chaotic systems can't be predicted because that computer has a finite precision in its numbers no matter how many decimals we choose to save and its in these lost decimal values that chaos springs up. Saturn's moons have chaotic orbits and a couple of moons have a dance where they switch orbits as they cross.

But my question is, as classical mechanics is completely predictable, shouldn't the Solar System be, too? So if we now predict the orbits for the next 100 million years and create a computer that can have 10 times more accuracy, we can extend the prediction. So if we would have a computer with bigger and bigger accuracy, would we be able to predict the position of each planet indefinitely?

 

[jedishrfu]

You're missing the point. You can't just use orbital mechanics to model the system. There are so many other unknowns that over time get magnified and alter whatever simulation you make. From state to state these changes may be infinitesimally small but over time they magnify and cause the real system to deviate from the modeled approximation.

 

[Silviu]

You're missing the point. You can't just use orbital mechanics to model the system. There are so many other unknowns that over time get magnified and alter whatever simulation you make. From state to state these changes may be infinitesimally small but over time they magnify and cause the real system to deviate from the modeled approximation.

I am not sure I understand. The only small effects I can think of, would be from GR (well of course there might be quantum fluctuations, but I assume they are insignificant for this purpose). So if you use GR instead of Newtonian Mechanics, shouldn't you get perfect results for a long period with a big enough computer accuracy?

 

[jedishrfu]

The sun radiates energy which pushes against the planets is not factored in. The sun is losing mass as it radiates energy. There are unpredictable solar storms which radiate even more energy. There's the loss of planetary rotational energy transferred to the moons... I'm sure there are other things some external influences... that I yet to hear about that would need to be considered.

https://en.wikipedia.org/wiki/Butterfly_effect

 

[anorlunda]

How about the influences of each asteroid, or each object in the Kuiper Belt, each comet, radiation pressure, solar wind, geomagnetic storms? You can't say negligably small if you want millions of orbits.

 

[Silviu]

How about the influences of each asteroid, or each object in the Kuiper Belt, each comet, radiation pressure, solar wind, geomagnetic storms? You can't say negligably small if you want millions of orbits.

Ok, maybe I was not clear enough. My question is not if it is feasible to simulate all of these, but if the Solar System is intrinsically deterministic. Like quantum mechanics can't be simulated perfectly because it is not deterministic (by this I mean the measurement, not the evolution of the wave function). Is the solar system deterministic (I am not asking if it can be simulated on a computer now)?

 

[anorlunda]

Ok, maybe I was not clear enough. My question is not if it is feasible to simulate all of these, but if the Solar System is intrinsically deterministic. Like quantum mechanics can't be simulated perfectly because it is not deterministic (by this I mean the measurement, not the evolution of the wave function). Is the solar system deterministic (I am not asking if it can be simulated on a computer now)?

In that case, re-read #4. Also read about chaos theory on Wikipedia if you're not familiar with that.